queueing theory: part 1

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This post is part of a series on Queueing Theory. The other articles can be found here:

A queueing system is a model with the following structure: customers arrive and join a queue to wait for service given by n servers. after receiving service, the customer exits the system. a fundamental result of queueing theory is little’s law.

theorem: for a queueing system in steady state, the average length of the queue is equivalent to the average arrival rate multiplied by the average waiting time. in other words,

L = λW

at amazon, i used little’s law all the time. in dynamic systems with n+ dependencies, it is very helpful to know where the bottlenecks of the system are and how to increase efficiencies, reduce time traps, and eliminate waste in order to increase material flow. in other words, we want product to flow as fast as possible: click-to-ship.

here’s an example:

say there’s a warehouse with 4000 pallets of product that turns ~4 times per year. do we have enough labor to support these transactions? using little’s law, we get

4,000 = λ(.25year)

so,

λ = 16,000 pallets/year

assuming a 10 hour shift per day of about 250 working days per year, there is roughly 2,000 working hours. this means, then,

λ = 8 pallets/hour

the analysis above is critically important to estimate the labor force required to move pallets, receive product, move product, and get work done, in general.

there are many more applications of queueing theory that i will explicate and share in the next while. queueing theory is a critical, underused, but very valuable principle in business.

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